Density

Sometimes it is necessary to estimate the density of a sample or coating from its chemical composition alone. This is especially true of quantitative depth profiling with GD-OES. Currently the most effective method is to calculate the unknown density r from the densities of pure materials, assuming constant atomic volume. That is(1)

where ci is the mass % of element i and ri the density of pure element i. For a proof of this equation, go further down on this page.

How good is this equation? Here is a selection of results:

Name

Description

Density(kg m-3)

Calc.(kg m-3)

Error(%)

Zn-Al

Zn50 Al50

3902

3997

0.4

Brass

Cu60 Zn30

8550

8327

-2.7

Stainless steel

Fe75 Cr12 Ni12

8010

7864

-1.8

Aluminium oxide

Al2O3

3980

2833

28.8

Silicon nitride

Si3N4

3300

2900

12.1

Acetaldehyde

CH3CHO

805

843

4.7

Generally the approach works well for metal alloys, typically giving results within a few percent of measured values. Non-metals, in particular oxides, do not fair as well, being in error by up to 40%. The assumption of constant volumes is clearly not working for these materials. The main reason lies with the nature of chemical bonds, the stronger the bond the closer the atoms are drawn together and the smaller the atomic volumes; some of the strongest bonds are in oxides. A secondary reason is in the arrangement of atoms, i.e. the crystal structure, since a different structure may have a different number of atoms per unit volume. So where do we go from here? A more refined model must include additional information about chemical bonds and crystal structure.

Correction for Oxides and Nitrides

Clearly the constant atomic volume equation is not accurate for oxides, it is also not accurate for other compounds with strong covalent bonds, such as nitrides, which reduce the apparent size of the atoms and hence increase the density.

To date, there is no general and easy method to include covalent bonds. But we know from the analysis whether O and N and other covalent bond forming elements are present. So we will assume that if they are present then they will form such bonds, and they will form them preferentially with those elements which give the lowest energy.

We will assume the order of oxide or nitride formation is determined by the electronegativities of the other elements present, beginning with the lowest. Hence, for example, Mg (1.31) will form an oxide before Al (1.61) and Al before Zn (1.65).

Then, as suggested by Analytis,(2) we simply look up a table of densities for known oxides and nitrides. To determine the density we then sum the specific densities (inverse densities) of the metals, oxides and nitrides, into the density equation above.

So now we can calculate densities using constant atomic volumes for up to five elements a-e, with oxide and nitride correction.

If, instead, you would like to measure density there is a simple method described by Jukka IsoPahkala of SP Swedish National Testing and Research Institute.

Prove of density equation:

Let's assume we have a binary alloy of elements a and b. One atom of a has a volume of Va and one of b has a volume Vb. The volume occupied by one mole of a is Va x NA, where NA is Avogadro's number (the number of atoms in a mole), and forb is Vb x NA. The mass of one mole of a is wa x NA, where wa is the atomic weight of a, and for b is wb x NA. The density ra of a is therefore

mass/volume = wa x NA / Va x NA = wa / Va,

and for b is wa / Va. Now suppose we mix A% of a with B% of b, and assume the atomic volumes do not change. The total volume of one mole is (A x Vb x NA + B x Vb x NA.)/ 100. The mass of one mole is (A x wa x NA + B x wb x NA) / 100. But instead of the density we will calculate the inverse density (the reason will become apparent)

This process can then be generalised to any number of elements with constant atomic volumes.